Why there is 2[itex]\pi[/itex] in every dirac delta function

in QF, every dirac delta function is accompanied by [itex]2\pi[/itex],i.e.[itex](2\pi)\delta(p-p_0)[/itex] or [itex](2\pi)^3\delta(\vec{p}-\vec{p_0})[/itex]

the intergral element in QF is [itex]\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_P}[/itex], it comes from the integral element [itex]\int\frac{d^4p}{(2\pi)^4}(2\pi)\delta(p^2-m^2)[/itex],I want to know why there is [itex]2\pi[/itex]in the second equation. Is it a convention? Is so where does the [itex]2\pi[/itex]comes from originally.

It's a convention, where you shift all the [itex]2 \pi[/itex] factors around in quantum mechanics or quantum field theory and where to put the - sign in the exponential. In the HEP community the most common convention is to put all the [itex]2\pi[/itex]'s to the momentum-space integrals, i.e., you define Green's functions etc. as

One representation of the delta function is
[tex]\delta(x)=\frac{1}{2\pi}\int dk e^{ikx}.[/tex]
If the integral appears in some equation without the [tex]\frac{1}{2\pi}[/tex],
you get [tex]2\pi\delta[/tex].